$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

Junjie Zhang; Shenzhou Zheng; Chunyan Zuo

doi:10.3934/dcdss.2021080

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September 2021, 14(9): 3305-3318. doi: 10.3934/dcdss.2021080

$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

Junjie Zhang ^1, , Shenzhou Zheng ^2,, and Chunyan Zuo ^3,

1.	School of Mathematical Sciences, Hebei Normal University, Shijiazhuang, Hebei 050016, China
2.	Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China
3.	School of Sciences, Hebei University of Science and Technology, Shijiazhuang, Hebei 050016, China

^* Corresponding author: Shenzhou Zheng

Received March 2020 Revised January 2021 Published September 2021 Early access June 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (No. 12001160), Natural Science Foundation of Hebei Province (No. A2019205218), Science Foundation of Hebei Normal University (No. L2019B02). The second author is supported by National Natural Science Foundation of China (No. 12071021)

We prove a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear elliptic equations $ F(x, u, Du, D^{2}u) = f(x) $ with oblique boundary condition in a bounded $ C^{2, \alpha} $-domain for every $ \alpha\in (0, 1) $. Here, the nonlinearities $ F $ is assumed to be asymptotically $ \delta $-regular to an operator $ G $ that is $ (\delta, R) $-vanishing with respect to $ x $. We employ the approach of constructing a regular problem by an appropriate transformation. With a similar argument, we also obtain a global $ W^{2, p} $-estimate for the viscosity solution to fully nonlinear parabolic equations $ F(x, t, u, Du, D^{2}u)-u_{t} = f(x, t) $ with oblique boundary condition in a bounded $ C^{3} $-domain.

Keywords: Fully nonlinear elliptic and parabolic equation, oblique boundary condition, asymptotically $ \delta $-regular, $ (\delta, R) $-vanishing condition, viscosity solution.

Mathematics Subject Classification: Primary: 35B65, 35J25, 35K20.

Citation: Junjie Zhang, Shenzhou Zheng, Chunyan Zuo. $ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values. Discrete and Continuous Dynamical Systems - S, 2021, 14 (9) : 3305-3318. doi: 10.3934/dcdss.2021080

References:

[1]	T. Alberico, C. Capozzoli, R. Schiattarella and L. D'Onofrio, G-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 129-137. doi: 10.3934/dcdss.2019009.
[2]	S.-S. Byun and J. Han, $W^{2, p}$-estimates for fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations, 268 (2020), 2125-2150. doi: 10.1016/j.jde.2019.09.018.
[3]	S.-S. Byun and J. Han, $L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, arXiv: 2012.07435v1 [math.AP], 43 pages.
[4]	S.-S. Byun, M. Lee and D. K. Palagachev, Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260 (2016), 4550-4571. doi: 10.1016/j.jde.2015.11.025.
[5]	S.-S. Byun, J. Oh and L. Wang, Global Calderón-Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN, 2015 (2015), 8289-8308. doi: 10.1093/imrn/rnu203.
[6]	L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213. doi: 10.2307/1971480.
[7]	L. A. Caffarelli and Q. Huang, Estimates in the generalized Campamato-John-Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., 118 (2003), 1-17. doi: 10.1215/S0012-7094-03-11811-6.
[8]	M. Chipot and L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 291-303. doi: 10.1017/S0308210500026378.
[9]	J. Choi, H. Dong and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374. doi: 10.3934/dcds.2018097.
[10]	H. Dong and N. V. Krylov, On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions, Commun. Partial Diff. Equ., 38 (2013), 1038-1068. doi: 10.1080/03605302.2012.756013.
[11]	H. Dong, N. V. Krylov and X. Li, On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains, St. Petersburg Math. J., 24 (2013), 39-69. doi: 10.1090/S1061-0022-2012-01231-8.
[12]	L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423. doi: 10.1512/iumj.1993.42.42019.
[13]	M. Foss, Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura Appl., 187 (2008), 263-321. doi: 10.1007/s10231-007-0045-2.
[14]	C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully nonlinear elliptic equations, Advances in Mathematics, 233 (2013), 196-206. doi: 10.1016/j.aim.2012.07.033.
[15]	N. V. Krylov, Linear and fully nonlinear elliptic equations with $L_{d}$-drift, Commun. Partial Diff. Equ., 45 (2020), 1778-1798. doi: 10.1080/03605302.2020.1805462.
[16]	T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427. doi: 10.1016/j.matpur.2012.02.004.
[17]	D. Li and K. Zhang, Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228 (2018), 923-967. doi: 10.1007/s00205-017-1209-x.
[18]	S. Liang and S. Zheng, Calderón-Zygmund estimate for asymptotically regular non-uniformly elliptic equations, J. Math. Anal. Appl., 484 (2020), 123749, 17 pp. doi: 10.1016/j.jmaa.2019.123749.
[19]	P. Marcellini, Regularity under general and $p, q$-growth conditions, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2009-2031. doi: 10.3934/dcdss.2020155.
[20]	E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Commun. Partial Diff. Equ., 31 (2006), 1227-1252. doi: 10.1080/03605300600634999.
[21]	C. Scheven and T. Schmidt, Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 469-507. doi: 10.2422/2036-2145.2009.3.04.
[22]	L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76. doi: 10.1002/cpa.3160450103.
[23]	N. Winter, $W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164. doi: 10.4171/ZAA/1377.
[24]	J. Zhang and S. Zheng, Lorentz estimates for asymptotically regular fully nonlinear parabolic equations, Math. Nachr., 291 (2018), 996-1008. doi: 10.1002/mana.201600497.
[25]	J. Zhang, M. Cai and S. Zheng, Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type, Nonlinear Anal., 180 (2019), 225-235. doi: 10.1016/j.na.2018.10.013.

show all references

References:

[1]	T. Alberico, C. Capozzoli, R. Schiattarella and L. D'Onofrio, G-convergence for non-divergence elliptic operators with VMO coefficients in $\mathbb{R}^3$, Discrete Contin. Dyn. Syst. Ser. S, 12 (2019), 129-137. doi: 10.3934/dcdss.2019009.
[2]	S.-S. Byun and J. Han, $W^{2, p}$-estimates for fully nonlinear elliptic equations with oblique boundary conditions, J. Differential Equations, 268 (2020), 2125-2150. doi: 10.1016/j.jde.2019.09.018.
[3]	S.-S. Byun and J. Han, $L^{p}$-estimates for the Hessians of solutions to fully nonlinear parabolic equations with oblique boundary conditions, arXiv: 2012.07435v1 [math.AP], 43 pages.
[4]	S.-S. Byun, M. Lee and D. K. Palagachev, Hessian estimates in weighted Lebesgue spaces for fully nonlinear elliptic equations, J. Differential Equations, 260 (2016), 4550-4571. doi: 10.1016/j.jde.2015.11.025.
[5]	S.-S. Byun, J. Oh and L. Wang, Global Calderón-Zygmund theory for asymptotically regular nonlinear elliptic and parabolic equations, Int. Math. Res. Not. IMRN, 2015 (2015), 8289-8308. doi: 10.1093/imrn/rnu203.
[6]	L. A. Caffarelli, Interior a priori estimates for solutions of fully nonlinear equations, Ann. of Math., 130 (1989), 189-213. doi: 10.2307/1971480.
[7]	L. A. Caffarelli and Q. Huang, Estimates in the generalized Campamato-John-Nirenberg spaces for fully nonlinear elliptic equations, Duke Math. J., 118 (2003), 1-17. doi: 10.1215/S0012-7094-03-11811-6.
[8]	M. Chipot and L. C. Evans, Linearisation at infinity and Lipschitz estimates for certain problems in the calculus of variations, Proc. Roy. Soc. Edinburgh Sect. A, 102 (1986), 291-303. doi: 10.1017/S0308210500026378.
[9]	J. Choi, H. Dong and D. Kim, Conormal derivative problems for stationary Stokes system in Sobolev spaces, Discrete Contin. Dyn. Syst., 38 (2018), 2349-2374. doi: 10.3934/dcds.2018097.
[10]	H. Dong and N. V. Krylov, On the existence of smooth solutions for fully nonlinear parabolic equations with measurable "coefficients" without convexity assumptions, Commun. Partial Diff. Equ., 38 (2013), 1038-1068. doi: 10.1080/03605302.2012.756013.
[11]	H. Dong, N. V. Krylov and X. Li, On fully nonlinear elliptic and parabolic equations with VMO coefficients in domains, St. Petersburg Math. J., 24 (2013), 39-69. doi: 10.1090/S1061-0022-2012-01231-8.
[12]	L. Escauriaza, $W^{2, n}$ a priori estimates for solutions to fully nonlinear equations, Indiana Univ. Math. J., 42 (1993), 413-423. doi: 10.1512/iumj.1993.42.42019.
[13]	M. Foss, Global regularity for almost minimizers of nonconvex variational problems, Ann. Mat. Pura Appl., 187 (2008), 263-321. doi: 10.1007/s10231-007-0045-2.
[14]	C. Imbert and L. Silvestre, $C^{1, \alpha}$ regularity of solutions of some degenerate fully nonlinear elliptic equations, Advances in Mathematics, 233 (2013), 196-206. doi: 10.1016/j.aim.2012.07.033.
[15]	N. V. Krylov, Linear and fully nonlinear elliptic equations with $L_{d}$-drift, Commun. Partial Diff. Equ., 45 (2020), 1778-1798. doi: 10.1080/03605302.2020.1805462.
[16]	T. Kuusi and G. Mingione, New perturbation methods for nonlinear parabolic problems, J. Math. Pures Appl., 98 (2012), 390-427. doi: 10.1016/j.matpur.2012.02.004.
[17]	D. Li and K. Zhang, Regularity for fully nonlinear elliptic equations with oblique boundary conditions, Arch. Ration. Mech. Anal., 228 (2018), 923-967. doi: 10.1007/s00205-017-1209-x.
[18]	S. Liang and S. Zheng, Calderón-Zygmund estimate for asymptotically regular non-uniformly elliptic equations, J. Math. Anal. Appl., 484 (2020), 123749, 17 pp. doi: 10.1016/j.jmaa.2019.123749.
[19]	P. Marcellini, Regularity under general and $p, q$-growth conditions, Discrete Contin. Dyn. Syst. Ser. S, 13 (2020), 2009-2031. doi: 10.3934/dcdss.2020155.
[20]	E. Milakis and L. E. Silvestre, Regularity for fully nonlinear elliptic equations with Neumann boundary data, Commun. Partial Diff. Equ., 31 (2006), 1227-1252. doi: 10.1080/03605300600634999.
[21]	C. Scheven and T. Schmidt, Asymptotically regular problems II: Partial Lipschitz continuity and a singular set of positive measure, Ann. Sc. Norm. Super. Pisa Cl. Sci., 8 (2009), 469-507. doi: 10.2422/2036-2145.2009.3.04.
[22]	L. Wang, On the regularity theory of fully nonlinear parabolic equations. I, Comm. Pure Appl. Math., 45 (1992), 27-76. doi: 10.1002/cpa.3160450103.
[23]	N. Winter, $W^{2, p}$ and $W^{1, p}$-estimates at the boundary for solutions of fully nonlinear, uniformly elliptic equations, Z. Anal. Anwend., 28 (2009), 129-164. doi: 10.4171/ZAA/1377.
[24]	J. Zhang and S. Zheng, Lorentz estimates for asymptotically regular fully nonlinear parabolic equations, Math. Nachr., 291 (2018), 996-1008. doi: 10.1002/mana.201600497.
[25]	J. Zhang, M. Cai and S. Zheng, Weighted Lorentz estimate for asymptotically regular parabolic equations of p(x, t)-Laplacian type, Nonlinear Anal., 180 (2019), 225-235. doi: 10.1016/j.na.2018.10.013.

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American Institute of Mathematical Sciences

$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

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$ W^{2, p} $-regularity for asymptotically regular fully nonlinear elliptic and parabolic equations with oblique boundary values

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